Abstract
We study the equilibrium distribution of relative strategy scores of agents in the asymmetric phase ($\alpha\equiv P/N\gtrsim 1$) of the basic Minority Game using sign-payoff, with $N$ agents holding two strategies over $P$ histories. We formulate a statistical model that makes use of the gauge freedom with respect to the ordering of an agent's strategies to quantify the correlation between the attendance and the distribution of strategies. The relative score $x\in\mathbb{Z}$ of the two strategies of an agent is described in terms of a one dimensional random walk with asymmetric jump probabilities, leading either to a static and asymmetric exponential distribution centered at $x=0$ for fickle agents or to diffusion with a positive or negative drift for frozen agents. In terms of scaled coordinates $x/\sqrt{N}$ and $t/N$ the distributions are uniquely given by $\alpha$ and in quantitative agreement with direct simulations of the game. As the model avoids the reformulation in terms of a constrained minimization problem it can be used for arbitrary payoff functions with little calculational effort and provides a transparent and simple formulation of the dynamics of the basic Minority Game in the asymmetric phase.
Highlights
A minority game can be exemplified by the following simple market analogy; An odd number N of traders must at each time step choose between two options, buying or selling a share, with the aim of picking the minority group
The Minority Game proposed by Challet and Zhang [2,3] formalizes this type of market dynamics where agents of limited intellect compete for a scarce resource by adapting to the aggregate input of all others [1,12]
In this paper we study the dynamics of the Minority Game in the asymmetric phase by formulating a simplified statistical model, focusing on finding probability distributions for the relative strategy scores
Summary
A minority game can be exemplified by the following simple market analogy; An odd number N of traders (agents) must at each time step choose between two options, buying or selling a share, with the aim of picking the minority group. Each agent has a set of strategies that, depending on the recent past history of minority groups going m time steps back, gives a prediction of the minority being buy or sell. The state space of the game is given by the strategy scores of each agent together with the recent history of minority groups, and the discrete time evolution in this space represents an intricate dynamical system. A somewhat oversimplified characterization of the dynamics is that the information about the last winning minority group for a given history gives a crowding effect [20] where many agents want to repeat the last winning outcome which counterproductively instead puts them in the majority group. In this paper we study the dynamics of the Minority Game in the asymmetric phase by formulating a simplified statistical model, focusing on finding probability distributions for the relative strategy scores. The MG is well understood from the classic works discussed above, it is our hope that the simplified model of the steady state attendance and score distributions presented in this paper provides an alternative and readily accessible perspective on this fascinating model
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